Context: Nowadays, the prediction of source code design problems plays an essential role in the software development industry, identifying defective architectural modules in advance. For this reason, some studies explored this subject in the last decade. Researchers and practitioners often need to create an overview of such studies considering the predictors of design problems, their pivotal contributions, the used prediction techniques, and research methods. Problem: However, the current literature lacks studies introducing a detailed mapping of published works. Objective: This article, therefore, aims at classifying the current literature, and pinpointing trends and challenges worth investigating in this research field. Method: We run a systematic mapping study following well-known guidelines. We applied a careful filtering process from a corpus of 894 candidate studies. In total, 35 primary studies were selected, analyzed, and categorized. Results: The main results are that a majority of the primary studies (1) explore Bloater bad smells, (2) use code complexity and size as predictors, (3) apply machine learning techniques to generate predictions, and (4) present a prediction proposal without an extensive empirical assessment. Conclusions: Predicting design problems is still in its infancy, showing plenty of room for future works. Finally, our findings can serve as a starting point for upcoming studies.